Hemimage temperaments: Difference between revisions

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Line 27: Line 27:
{{Val list|legend=1| 58, 118, 294, 412d, 530d }}
{{Val list|legend=1| 58, 118, 294, 412d, 530d }}


[[Badness]]: 0.0843
[[Badness]]: 0.084317


== 11-limit ==
== 11-limit ==
Line 40: Line 40:
Vals: {{Val list| 58, 118, 294, 412d }}
Vals: {{Val list| 58, 118, 294, 412d }}


Badness: 0.0305
Badness: 0.030461


= Chromat =
= Chromat =
Line 57: Line 57:
{{Val list|legend=1| 60, 99, 258, 357, 456 }}
{{Val list|legend=1| 60, 99, 258, 357, 456 }}


[[Badness]]: 0.0575
[[Badness]]: 0.057499


= Degrees =
= Degrees =
Line 72: Line 72:
{{Val list|legend=1| 60, 80, 140, 640b, 780b, 920b }}
{{Val list|legend=1| 60, 80, 140, 640b, 780b, 920b }}


[[Badness]]: 0.1065
[[Badness]]: 0.106471


== 11-limit ==
== 11-limit ==
Line 85: Line 85:
Vals: {{Val list| 80, 140, 360, 500be, 860bde }}
Vals: {{Val list| 80, 140, 360, 500be, 860bde }}


Badness: 0.0468
Badness: 0.046770


== 13-limit ==
== 13-limit ==
Line 98: Line 98:
Vals: {{Val list| 60e, 80, 140, 500be, 640be, 780be }}
Vals: {{Val list| 60e, 80, 140, 500be, 640be, 780be }}


Badness: 0.0327
Badness: 0.032718


= Subfourth =
= Subfourth =
Line 106: Line 106:


[[Mapping]]: [{{val| 1 0 17 4 }}, {{val| 0 4 -37 -3 }}]
[[Mapping]]: [{{val| 1 0 17 4 }}, {{val| 0 4 -37 -3 }}]
{{Multival|legend=1| 4 -37 -3 -68 -16 97 }}


[[POTE generator]]: ~21/16 = 475.991
[[POTE generator]]: ~21/16 = 475.991
Line 111: Line 113:
{{Val list|legend=1| 58, 121, 179, 300bd, 479bcd }}
{{Val list|legend=1| 58, 121, 179, 300bd, 479bcd }}


[[Badness]]: 0.1407
[[Badness]]: 0.140722


== 11-limit ==
== 11-limit ==
Line 124: Line 126:
Vals: {{Val list| 58, 121, 179e, 300bde }}
Vals: {{Val list| 58, 121, 179e, 300bde }}


Badness: 0.0453
Badness: 0.045323


== 13-limit ==
== 13-limit ==
Line 137: Line 139:
Vals: {{Val list| 58, 121, 179ef, 300bdef }}
Vals: {{Val list| 58, 121, 179ef, 300bdef }}


Badness: 0.0238
Badness: 0.023800


= Bisupermajor =
= Bisupermajor =
Line 147: Line 149:


[[Mapping]]: [{{val| 2 1 6 1 }}, {{val| 0 8 -5 17 }}]
[[Mapping]]: [{{val| 2 1 6 1 }}, {{val| 0 8 -5 17 }}]
{{Multival|legend=1| 16 -10 34 -53 9 107 }}


[[POTE generator]]: ~192/175 = 162.8061
[[POTE generator]]: ~192/175 = 162.8061
Line 152: Line 156:
{{Val list|legend=1| 22, 74d, 96d, 118, 140, 258, 398 }}
{{Val list|legend=1| 22, 74d, 96d, 118, 140, 258, 398 }}


[[Badness]]: 0.0655
[[Badness]]: 0.065492


== 11-limit ==
== 11-limit ==
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 166: Line 169:
Vals: {{Val list| 22, 74d, 96d, 118, 258e, 376de }}
Vals: {{Val list| 22, 74d, 96d, 118, 258e, 376de }}


Badness: 0.0321
Badness: 0.032080
 
= Cotoneum =
The ''cotoneum'' temperament (41&217, named by [[User:Xenllium|Xenllium]] after the Latin for "quince") tempers out the [[Quince clan|quince comma]], 823543/819200 and the [[garischisma]], 33554432/33480783. This temperament is supported by [[41edo|41]], [[176edo|176]], [[217edo|217]], and [[258edo|258]] EDOs, and can be extended to the 11-, 13-, 17-, and 19-limit by adding 441/440, 364/363, 595/594, and 343/342 to the comma list in this order.
 
Subgroup: 2.3.5.7
 
[[Comma list]]: 10976/10935, 823543/819200
 
[[Mapping]]: [{{val|1 2 -18 -3}}, {{val|0 -1 49 14}}]
 
{{Multival|legend=1| 1 -49 -14 -80 -25 105 }}
 
[[POTE generator]]: ~3/2 = 702.317
 
{{Val list|legend=1| 41, 135c, 176, 217, 258, 475 }}
 
[[Badness]]: 0.105632
 
== 11-limit ==
Subgroup: 2.3.5.7.11
 
Comma list: 441/440, 10976/10935, 16384/16335
 
Mapping: [{{val|1 2 -18 -3 13}}, {{val|0 -1 49 14 -23}}]
 
POTE generator: ~3/2 = 702.303
 
Vals: {{Val list| 41, 176, 217 }}
 
Badness: 0.050966
 
== 13-limit ==
Subgroup: 2.3.5.7.11.13
 
Comma list: 364/363, 441/440, 3584/3575, 10976/10935
 
Mapping: [{{val|1 2 -18 -3 13 29}}, {{val|0 -1 49 14 -23 -61}}]
 
POTE generator: ~3/2 = 702.306
 
Vals: {{Val list| 41, 176, 217 }}
 
Badness: 0.036951
 
== 17-limit ==
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 364/363, 441/440, 595/594, 3584/3575, 8281/8262
 
Mapping: [{{val|1 2 -18 -3 13 29 41}}, {{val|0 -1 49 14 -23 -61 -89}}]
 
POTE generator: ~3/2 = 702.307
 
Vals: {{Val list| 41, 176, 217 }}
 
Badness: 0.029495
 
== 19-limit ==
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 343/342, 364/363, 441/440, 595/594, 1216/1215, 1729/1728
 
Mapping: [{{val|1 2 -18 -3 13 29 41 -14}}, {{val|0 -1 49 14 -23 -61 -89 44}}]
 
POTE generator: ~3/2 = 702.308
 
Vals: {{Val list| 41, 176, 217 }}
 
Badness: 0.021811


[[Category:Regular temperament theory]]
[[Category:Regular temperament theory]]

Revision as of 13:01, 11 May 2021

Hemimage temperaments temper out the hemimage comma, [5 -7 -1 3⟩ = 10976/10935.

Discussed elsewhere are:

Commatic

Subgroup: 2.3.5.7

Comma list: 10976/10935, 50421/50000

Mapping: [⟨2 3 4 5], ⟨0 5 19 18]]

Wedgie: ⟨⟨ 10 38 36 37 29 -23 ]]

POTE generator: ~81/80 = 20.377

Template:Val list

Badness: 0.084317

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 3388/3375, 8019/8000

Mapping: [⟨2 3 4 5 6], ⟨0 5 19 18 27]]

POTE generator: ~81/80 = 20.390

Vals: Template:Val list

Badness: 0.030461

Chromat

See also: Amity family

Subgroup: 2.3.5.7

Comma list: 10976/10935, 235298/234375

Mapping: [⟨3 4 5 6], ⟨0 5 13 16]]

Wedgie: ⟨⟨ 15 39 48 27 34 2 ]]

POTE generator: ~28/27 = 60.528

Template:Val list

Badness: 0.057499

Degrees

Subgroup: 2.3.5.7

Comma list: 10976/10935, 390625/388962

Mapping: [⟨20 0 -17 -39], ⟨0 1 2 3]]

Wedgie: ⟨⟨ 20 40 60 17 39 27 ]]

POTE generator: ~3/2 = 703.015

Template:Val list

Badness: 0.106471

11-limit

Subgroup: 2.3.5.7.11

Comma list: 1331/1323, 1375/1372, 2200/2187

Mapping: [⟨20 0 -17 -39 -26], ⟨0 1 2 3 3]]

POTE generator: ~3/2 = 703.231

Vals: Template:Val list

Badness: 0.046770

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 352/351, 1001/1000, 1331/1323

Mapping: [⟨20 0 -17 -39 -26 74], ⟨0 1 2 3 3 0]]

POTE generator: ~3/2 = 703.080

Vals: Template:Val list

Badness: 0.032718

Subfourth

Subgroup: 2.3.5.7

Comma list: 10976/10935, 65536/64827

Mapping: [⟨1 0 17 4], ⟨0 4 -37 -3]]

Wedgie: ⟨⟨ 4 -37 -3 -68 -16 97 ]]

POTE generator: ~21/16 = 475.991

Template:Val list

Badness: 0.140722

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 896/891, 12005/11979

Mapping: [⟨1 0 17 4 11], ⟨0 4 -37 -3 -19]]

POTE generator: ~21/16 = 475.995

Vals: Template:Val list

Badness: 0.045323

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 364/363, 540/539, 676/675

Mapping: [⟨1 0 17 4 11 16], ⟨0 4 -37 -3 -19 -31]]

POTE generator: ~21/16 = 475.996

Vals: Template:Val list

Badness: 0.023800

Bisupermajor

See also: Very high accuracy temperaments § Kwazy

Subgroup: 2.3.5.7

Comma list: 10976/10935, 65625/65536

Mapping: [⟨2 1 6 1], ⟨0 8 -5 17]]

Wedgie: ⟨⟨ 16 -10 34 -53 9 107 ]]

POTE generator: ~192/175 = 162.8061

Template:Val list

Badness: 0.065492

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 3388/3375, 9801/9800

Mapping: [⟨2 1 6 1 8], ⟨0 8 -5 17 -4]]

POTE generators: ~11/10 = 162.7733

Vals: Template:Val list

Badness: 0.032080

Cotoneum

The cotoneum temperament (41&217, named by Xenllium after the Latin for "quince") tempers out the quince comma, 823543/819200 and the garischisma, 33554432/33480783. This temperament is supported by 41, 176, 217, and 258 EDOs, and can be extended to the 11-, 13-, 17-, and 19-limit by adding 441/440, 364/363, 595/594, and 343/342 to the comma list in this order.

Subgroup: 2.3.5.7

Comma list: 10976/10935, 823543/819200

Mapping: [⟨1 2 -18 -3], ⟨0 -1 49 14]]

Wedgie: ⟨⟨ 1 -49 -14 -80 -25 105 ]]

POTE generator: ~3/2 = 702.317

Template:Val list

Badness: 0.105632

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 10976/10935, 16384/16335

Mapping: [⟨1 2 -18 -3 13], ⟨0 -1 49 14 -23]]

POTE generator: ~3/2 = 702.303

Vals: Template:Val list

Badness: 0.050966

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 441/440, 3584/3575, 10976/10935

Mapping: [⟨1 2 -18 -3 13 29], ⟨0 -1 49 14 -23 -61]]

POTE generator: ~3/2 = 702.306

Vals: Template:Val list

Badness: 0.036951

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 364/363, 441/440, 595/594, 3584/3575, 8281/8262

Mapping: [⟨1 2 -18 -3 13 29 41], ⟨0 -1 49 14 -23 -61 -89]]

POTE generator: ~3/2 = 702.307

Vals: Template:Val list

Badness: 0.029495

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 343/342, 364/363, 441/440, 595/594, 1216/1215, 1729/1728

Mapping: [⟨1 2 -18 -3 13 29 41 -14], ⟨0 -1 49 14 -23 -61 -89 44]]

POTE generator: ~3/2 = 702.308

Vals: Template:Val list

Badness: 0.021811

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